Stability and natural frequencies of a weakened Timoshenko beam-column with generalized end conditions under constant axial load

The stability and free vibration analyses of a Timoshenko beam-column with generalized end conditions (i.e., with semi-rigid flexural connections and lateral bracings at both ends) subjected to constant axial load (tension or compression), and weakened by a cracked section along its span are presented. The magnitude and location of the weakened section are both arbitrary and independent of each other. The magnitude of the crack is modeled as an intermediate flexural connection of zero length producing a member with two-segments with rotational discontinuity at the weakened section but of identical lateral deflection. The proposed model offers the option of considering the beneficial effects of an additional lateral bracing located at the weakened section to alleviate the detrimental effects of the rotational discontinuity on the stability and natural frequencies of the whole beam-column. The proposed model includes the following coupling effects: (1) shear and bending deformations along the member's span; (2) the translational and rotational masses of the member uniformly distributed along its span; (3) constant axial load (tension or compression) applied at both ends; and (4) the shear forces along the member induced by the applied axial load as the beam-column deforms according to two different approaches: proportional to the bending rotation or to the total slope of the member axis. A flowchart is included that shows the steps necessary to carry out the stability and free vibration analyses. Finally, to show the validity and simplicity of the proposed method and equations five comprehensive examples are presented using the two approaches of the induced shear force and the obtained results are compared to those calculated by other analytical methods including the finite element method.